The big red OUT sign that Cummings draws casually on the wall of his office is straighten out later on with letters of the same length and width.
Cummings is found typing on his laptop in a storeroom completely in the dark. There should be some illumination from the laptop's monitor.
Boris Johnson remarks on the use of the real NHS logo on the side of the bus, and asks if they're allowed to use it. But what's shown isn't the real NHS logo - the real one is italic, whereas the one show is not - presumably since the film-makers weren't allowed to use the real one.
The film shows the referendum results being announced in a grand revealing the following morning in Manchester. In practice, the same districts as local elections were used for logistical reasons and so the result came in in dribs and drabs, announced on the television constituency by constituency. The result was that the final decision to Leave was known long before all the votes were counted.
UKIP MP Douglas Carswell is portrayed as being unfamiliar with the Jaywick area of his constituency (electoral district) and is accused by Dominic Cummings of having ignored it. In fact, Carswell knew the area and as an MP before the referendum was involved in a local campaign of trying to prevent TV companies from filming there to depict it in a negative light.
Cummings refers to "The Art of War" being written by "Tzu." "The Art of War" was written by Sun Tzu, which literally translates to "Master Sun"; therefore, it would have been by "Sun," if only one name is to be used, because "Tzu" is a title, not an actual name.